Properties

Label 1089.3.b.e
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,3,Mod(485,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.485");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 2 q^{4} + 2 \beta_{3} q^{5} - 7 \beta_{2} q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 2 q^{4} + 2 \beta_{3} q^{5} - 7 \beta_{2} q^{7} - 2 \beta_1 q^{8} - 12 \beta_{2} q^{10} - 2 \beta_{2} q^{13} - 7 \beta_{3} q^{14} - 20 q^{16} - 10 \beta_1 q^{17} - \beta_{2} q^{19} - 4 \beta_{3} q^{20} - \beta_{3} q^{23} - 47 q^{25} - 2 \beta_{3} q^{26} + 14 \beta_{2} q^{28} + 13 \beta_1 q^{29} + 37 q^{31} + 12 \beta_1 q^{32} - 60 q^{34} + 42 \beta_1 q^{35} + 61 q^{37} - \beta_{3} q^{38} - 24 \beta_{2} q^{40} - 17 \beta_1 q^{41} - 26 \beta_{2} q^{43} + 6 \beta_{2} q^{46} + 7 \beta_{3} q^{47} + 98 q^{49} + 47 \beta_1 q^{50} + 4 \beta_{2} q^{52} + \beta_{3} q^{53} - 14 \beta_{3} q^{56} + 78 q^{58} - 5 \beta_{3} q^{59} + 9 \beta_{2} q^{61} - 37 \beta_1 q^{62} - 8 q^{64} + 12 \beta_1 q^{65} - 41 q^{67} + 20 \beta_1 q^{68} + 252 q^{70} + 18 \beta_{3} q^{71} + 35 \beta_{2} q^{73} - 61 \beta_1 q^{74} + 2 \beta_{2} q^{76} - 75 \beta_{2} q^{79} - 40 \beta_{3} q^{80} - 102 q^{82} + 39 \beta_1 q^{83} - 120 \beta_{2} q^{85} - 26 \beta_{3} q^{86} - 12 \beta_{3} q^{89} + 42 q^{91} + 2 \beta_{3} q^{92} - 42 \beta_{2} q^{94} + 6 \beta_1 q^{95} + 179 q^{97} - 98 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 80 q^{16} - 188 q^{25} + 148 q^{31} - 240 q^{34} + 244 q^{37} + 392 q^{49} + 312 q^{58} - 32 q^{64} - 164 q^{67} + 1008 q^{70} - 408 q^{82} + 168 q^{91} + 716 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{3} - 9\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} - 9\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
0.517638i
1.93185i
1.93185i
0.517638i
2.44949i 0 −2.00000 8.48528i 0 −12.1244 4.89898i 0 −20.7846
485.2 2.44949i 0 −2.00000 8.48528i 0 12.1244 4.89898i 0 20.7846
485.3 2.44949i 0 −2.00000 8.48528i 0 12.1244 4.89898i 0 20.7846
485.4 2.44949i 0 −2.00000 8.48528i 0 −12.1244 4.89898i 0 −20.7846
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.e 4
3.b odd 2 1 inner 1089.3.b.e 4
11.b odd 2 1 inner 1089.3.b.e 4
33.d even 2 1 inner 1089.3.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.e 4 1.a even 1 1 trivial
1089.3.b.e 4 3.b odd 2 1 inner
1089.3.b.e 4 11.b odd 2 1 inner
1089.3.b.e 4 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 147 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1014)^{2} \) Copy content Toggle raw display
$31$ \( (T - 37)^{4} \) Copy content Toggle raw display
$37$ \( (T - 61)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1734)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2028)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 882)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 450)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 243)^{2} \) Copy content Toggle raw display
$67$ \( (T + 41)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 5832)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 3675)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 16875)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9126)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$97$ \( (T - 179)^{4} \) Copy content Toggle raw display
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